Newspace parameters
Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 273.bz (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.17991597518\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −2.19079 | − | 0.587020i | −0.866025 | − | 0.500000i | 2.72291 | + | 1.57208i | −0.477944 | + | 1.78371i | 1.60377 | + | 1.60377i | 0.565048 | − | 2.58471i | −1.83495 | − | 1.83495i | 0.500000 | + | 0.866025i | 2.09415 | − | 3.62718i |
31.2 | −2.05139 | − | 0.549668i | −0.866025 | − | 0.500000i | 2.17401 | + | 1.25516i | 1.02157 | − | 3.81253i | 1.50172 | + | 1.50172i | 2.63560 | − | 0.231538i | −0.766371 | − | 0.766371i | 0.500000 | + | 0.866025i | −4.19125 | + | 7.25946i |
31.3 | −1.27474 | − | 0.341564i | −0.866025 | − | 0.500000i | −0.223767 | − | 0.129192i | −0.0726797 | + | 0.271244i | 0.933171 | + | 0.933171i | −1.12096 | + | 2.39655i | 2.10746 | + | 2.10746i | 0.500000 | + | 0.866025i | 0.185295 | − | 0.320940i |
31.4 | −0.812358 | − | 0.217671i | −0.866025 | − | 0.500000i | −1.11951 | − | 0.646347i | 0.429239 | − | 1.60194i | 0.594687 | + | 0.594687i | −2.64477 | + | 0.0720542i | 1.95812 | + | 1.95812i | 0.500000 | + | 0.866025i | −0.697391 | + | 1.20792i |
31.5 | −0.227099 | − | 0.0608509i | −0.866025 | − | 0.500000i | −1.68418 | − | 0.972362i | −0.890829 | + | 3.32462i | 0.166248 | + | 0.166248i | 1.43872 | − | 2.22038i | 0.655801 | + | 0.655801i | 0.500000 | + | 0.866025i | 0.404612 | − | 0.700809i |
31.6 | 0.905729 | + | 0.242689i | −0.866025 | − | 0.500000i | −0.970604 | − | 0.560379i | −0.812951 | + | 3.03397i | −0.663040 | − | 0.663040i | −0.856042 | + | 2.50344i | −2.06919 | − | 2.06919i | 0.500000 | + | 0.866025i | −1.47263 | + | 2.55066i |
31.7 | 0.918377 | + | 0.246078i | −0.866025 | − | 0.500000i | −0.949189 | − | 0.548015i | 0.213650 | − | 0.797354i | −0.672298 | − | 0.672298i | −1.42430 | − | 2.22965i | −2.08146 | − | 2.08146i | 0.500000 | + | 0.866025i | 0.392423 | − | 0.679697i |
31.8 | 2.20302 | + | 0.590298i | −0.866025 | − | 0.500000i | 2.77280 | + | 1.60088i | −0.460631 | + | 1.71910i | −1.61272 | − | 1.61272i | 2.30189 | + | 1.30433i | 1.93810 | + | 1.93810i | 0.500000 | + | 0.866025i | −2.02956 | + | 3.51530i |
31.9 | 2.52924 | + | 0.677708i | −0.866025 | − | 0.500000i | 4.20573 | + | 2.42818i | 1.05058 | − | 3.92082i | −1.85153 | − | 1.85153i | −2.39518 | + | 1.12389i | 5.28864 | + | 5.28864i | 0.500000 | + | 0.866025i | 5.31435 | − | 9.20472i |
73.1 | −0.566824 | + | 2.11542i | 0.866025 | + | 0.500000i | −2.42164 | − | 1.39814i | 0.631372 | + | 0.169176i | −1.54859 | + | 1.54859i | 1.92845 | + | 1.81137i | 1.23310 | − | 1.23310i | 0.500000 | + | 0.866025i | −0.715753 | + | 1.23972i |
73.2 | −0.517606 | + | 1.93173i | 0.866025 | + | 0.500000i | −1.73162 | − | 0.999754i | −3.58432 | − | 0.960415i | −1.41413 | + | 1.41413i | −2.59127 | + | 0.534135i | −0.000695828 | 0 | 0.000695828i | 0.500000 | + | 0.866025i | 3.71053 | − | 6.42683i |
73.3 | −0.458633 | + | 1.71164i | 0.866025 | + | 0.500000i | −0.987324 | − | 0.570032i | 2.18934 | + | 0.586631i | −1.25301 | + | 1.25301i | −0.537086 | − | 2.59066i | −1.07751 | + | 1.07751i | 0.500000 | + | 0.866025i | −2.00820 | + | 3.47831i |
73.4 | −0.164102 | + | 0.612438i | 0.866025 | + | 0.500000i | 1.38390 | + | 0.798995i | 2.57523 | + | 0.690032i | −0.448336 | + | 0.448336i | −1.84289 | + | 1.89836i | −1.61311 | + | 1.61311i | 0.500000 | + | 0.866025i | −0.845203 | + | 1.46393i |
73.5 | 0.0455765 | − | 0.170094i | 0.866025 | + | 0.500000i | 1.70520 | + | 0.984495i | −0.317778 | − | 0.0851485i | 0.124517 | − | 0.124517i | 1.64846 | − | 2.06944i | 0.494209 | − | 0.494209i | 0.500000 | + | 0.866025i | −0.0289665 | + | 0.0501714i |
73.6 | 0.0704795 | − | 0.263033i | 0.866025 | + | 0.500000i | 1.66783 | + | 0.962923i | −2.42785 | − | 0.650540i | 0.192554 | − | 0.192554i | 1.21724 | + | 2.34911i | 0.755936 | − | 0.755936i | 0.500000 | + | 0.866025i | −0.342227 | + | 0.592755i |
73.7 | 0.345245 | − | 1.28847i | 0.866025 | + | 0.500000i | 0.191081 | + | 0.110321i | 1.14557 | + | 0.306956i | 0.943228 | − | 0.943228i | −2.38403 | − | 1.14734i | 2.09457 | − | 2.09457i | 0.500000 | + | 0.866025i | 0.791009 | − | 1.37007i |
73.8 | 0.542736 | − | 2.02552i | 0.866025 | + | 0.500000i | −2.07611 | − | 1.19864i | 0.821926 | + | 0.220234i | 1.48278 | − | 1.48278i | 2.59841 | − | 0.498246i | −0.589092 | + | 0.589092i | 0.500000 | + | 0.866025i | 0.892178 | − | 1.54530i |
73.9 | 0.703128 | − | 2.62411i | 0.866025 | + | 0.500000i | −4.65951 | − | 2.69017i | −1.03350 | − | 0.276925i | 1.92098 | − | 1.92098i | −1.53729 | − | 2.15331i | −6.49357 | + | 6.49357i | 0.500000 | + | 0.866025i | −1.45336 | + | 2.51730i |
187.1 | −0.566824 | − | 2.11542i | 0.866025 | − | 0.500000i | −2.42164 | + | 1.39814i | 0.631372 | − | 0.169176i | −1.54859 | − | 1.54859i | 1.92845 | − | 1.81137i | 1.23310 | + | 1.23310i | 0.500000 | − | 0.866025i | −0.715753 | − | 1.23972i |
187.2 | −0.517606 | − | 1.93173i | 0.866025 | − | 0.500000i | −1.73162 | + | 0.999754i | −3.58432 | + | 0.960415i | −1.41413 | − | 1.41413i | −2.59127 | − | 0.534135i | −0.000695828 | 0 | 0.000695828i | 0.500000 | − | 0.866025i | 3.71053 | + | 6.42683i |
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.bb | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 273.2.bz.a | ✓ | 36 |
3.b | odd | 2 | 1 | 819.2.fn.f | 36 | ||
7.d | odd | 6 | 1 | 273.2.bz.b | yes | 36 | |
13.d | odd | 4 | 1 | 273.2.bz.b | yes | 36 | |
21.g | even | 6 | 1 | 819.2.fn.g | 36 | ||
39.f | even | 4 | 1 | 819.2.fn.g | 36 | ||
91.bb | even | 12 | 1 | inner | 273.2.bz.a | ✓ | 36 |
273.cb | odd | 12 | 1 | 819.2.fn.f | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
273.2.bz.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
273.2.bz.a | ✓ | 36 | 91.bb | even | 12 | 1 | inner |
273.2.bz.b | yes | 36 | 7.d | odd | 6 | 1 | |
273.2.bz.b | yes | 36 | 13.d | odd | 4 | 1 | |
819.2.fn.f | 36 | 3.b | odd | 2 | 1 | ||
819.2.fn.f | 36 | 273.cb | odd | 12 | 1 | ||
819.2.fn.g | 36 | 21.g | even | 6 | 1 | ||
819.2.fn.g | 36 | 39.f | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{36} - 63 T_{2}^{32} - 44 T_{2}^{31} + 284 T_{2}^{29} + 3022 T_{2}^{28} + 2452 T_{2}^{27} + 452 T_{2}^{26} - 8954 T_{2}^{25} - 63405 T_{2}^{24} - 57946 T_{2}^{23} - 17820 T_{2}^{22} + 177534 T_{2}^{21} + 981706 T_{2}^{20} + \cdots + 144 \)
acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).